On the computation of all eigenvalues for the eigenvalue complementarity problem
In this paper, a parametric algorithm is introduced for computing all eigenvalues for two Eigenvalue Complementarity Problems discussed in the literature. The algorithm searches a finite number of nested intervals <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$[\bar{l}, \bar{u}]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mover accent="true"> <mrow> <mi>l</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mover accent="true"> <mrow> <mi>u</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation> in such a way that, in each iteration, either an eigenvalue is computed in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$[\bar{l}, \bar{u}]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mover accent="true"> <mrow> <mi>l</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mover accent="true"> <mrow> <mi>u</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation> or a certificate of nonexistence of an eigenvalue in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$[\bar{l}, \bar{u}]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mover accent="true"> <mrow> <mi>l</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mover accent="true"> <mrow> <mi>u</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation> is provided. A hybrid method that combines an enumerative method [<CitationRef CitationID="CR1">1</CitationRef>] and a semi-smooth algorithm [<CitationRef CitationID="CR2">2</CitationRef>] is discussed for dealing with the Eigenvalue Complementarity Problem over an interval <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$[\bar{l}, \bar{u}]$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">[</mo> <mover accent="true"> <mrow> <mi>l</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mover accent="true"> <mrow> <mi>u</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo stretchy="false">]</mo> </mrow> </math> </EquationSource> </InlineEquation>. Computational experience is presented to illustrate the efficacy and efficiency of the proposed techniques. Copyright Springer Science+Business Media New York 2014
